Shadow cast by Kerr-like black hole in the presence of plasma in Einstein-bumblebee gravity

In the Einstein-bumblebee gravity, the Lorentz symmetry is spontaneously broken by a vector field. In this paper, we attempt to test the Lorentz symmetry via the observation of the shadow cast by the Kerr-like black hole with or without plasma. A novel phenomenon of the Lorentz-violating parameter on the shadow is observed. The result shows that when the observer gradually moves from the poles to the equatorial plane, the shadow radius $R_{rm s}$ firstly decreases and then increases with the Lorentz-violating parameter. Such nonmonotonic behavior provides us an important understanding on the black hole shadow in the Einstein-bumblebee gravity. Besides, three more distortion observables are calculated, and found to increase with the Lorentz-violating parameter. Moreover, when a homogeneous plasma is present, the motion of the photon is analyzed. We further observe that the refractive index shrinks the size, while enhances the deformation of the shadow. Finally, adopting the observed data of the diameter of M87$^*$, we find the refractive index is more favored in (0.914, 1).